Let $W$ be a Weyl group with root system $R$ and with set of positive roots $R^+$. Let $\tilde{R}^+$ be the set of $B$-cosmall roots, i.e. positive roots $\alpha$ which satisfy $\ell(s_\alpha)=2\operatorname{ht}(\alpha^\vee)-1$, where $\ell$ denotes the length function and where $\operatorname{ht}$ denotes the obvious height function. Let $w_o$ be the longest element of $W$. Let $Q^\vee$ be the coroot lattice.
For a sequence of positive roots $\gamma_1,\ldots,\gamma_\ell$, define $$ \delta_i(\gamma_1,\ldots,\gamma_\ell)=\begin{cases} \hphantom{-}1\,,&\text{if }\gamma_i\in\tilde{R}^+\text{ and }\ell(s_{\gamma_1}\cdots s_{\gamma_i})=\ell(s_{\gamma_1}\cdots s_{\gamma_{i-1}})+\ell(s_{\gamma_i})\,,\\ \hphantom{-}0\,,&\text{if }\ell(s_{\gamma_1}\cdots s_{\gamma_i})=\ell(s_{\gamma_1}\cdots s_{\gamma_{i-1}})-1\,,\\ -1\,,&\text{otherwise}\,. \end{cases} $$
We say that an effective element $d\in Q^\vee$ satisfies Property $(\mathrm{A})$ if the following condition is satisfied: There exists a sequence of positive roots $\gamma_1,\ldots,\gamma_\ell$ and a sequence of simple roots $\beta_1,\ldots,\beta_\ell$ such that
- $w_o=s_{\gamma_\ell}\cdots s_{\gamma_1}$,
- $s_{\beta_1}\cdots s_{\beta_\ell}$ is a reduced word,
- $\beta_i$ is in the support of $\gamma_i$ for all $1\leq i\leq\ell$,
- $\delta_i(\gamma_1,\ldots,\gamma_\ell)\in\{0,1\}$ for all $1\leq i\leq\ell$,
- $\left(\sum_{j=1}^i\delta_j(\gamma_1,\ldots,\gamma_\ell)\gamma_j^\vee,\beta\right)\leq 2$ for all simple roots $\beta$ and for all $1\leq i\leq\ell$, and such that
- $d=\sum_{i=1}^\ell\delta_i(\gamma_1,\ldots,\gamma_\ell)\gamma_i^\vee$.
QUESTION 1. Can you compute the elements of the set $$ \{d\in Q^\vee\text{ effective with Property $(\mathrm{A})$}\} $$ which are maximal with respect to the natural partial order on $Q^\vee$?
QUESTION 2 (maybe simpler). Can you compute the number $$ \mathscr{D}_R=\operatorname{max}\{\operatorname{ht}(d)\mid d\in Q^\vee\text{ effective with Property $(\mathrm{A})$}\}\,? $$
REMARKS. If you can answer Question 2, it would be completely sufficient for me. I posed Question 1 only out of curiosity. If you can provide an implementation in SageMath or similar software which provides the number in Question 2 for type $\mathsf{A}_n$ and given $n$, I would be more than happy. I cannot do this for myself, unfortunately, because I am not experienced enough with computer algebra software.
I thank everyone for their help!
EDIT. The expected outcome for type $\mathsf{A}_n$ is as follows: $$ \mathscr{D}_{R\text{ of type $\mathsf{A}_n$}}=\begin{cases} \left[\tfrac{n}{2}\right]\left(\left[\tfrac{n}{2}\right]+1\right)\,,&\text{if $n$ is even}\,,\\ \left(\left[\tfrac{n}{2}\right]+1\right)^2\,,&\text{if $n$ is odd}\,, \end{cases} $$ where $[-]$ means the floor function. Can you confirm or disprove this expected outcome?
